Two other measurements, the median and the mode, are often used to define distribution. The median, or middle item, is helpful for establishing the center of the data: it halves the number of data items. The median has the advantage of discounting extreme values, which might distort the arithmetic mean. The mode is the most commonly occurring value in Figure 2-3 the mode is the highest point.

In a normally distributed price series, the mean, median, and mode will all occur at the same value; however, as the data become skewed, these values will move farther apart. The general relationship is

FIGURE 2-3 Hypothetical price distribution d;ewed to the right, showing the relationship of the mode, median, and mean.

mean := median > mode

The mean, median, and mode help to tell whether data is normally distributed or d;ewed. A normal distribution IS commonly called a bell curve, and values fall equally on both sides of the mean. For much of the work done with price and performance data, the distributions tend to extend out toward the right (positive values) and be more cut off on the left (negative values). If you were to chart a distribution of trading profits and losses based on a trend sj-stem with a fixed stop-loss, you would get profits that could range from zero to very laie values, while the losses would be theoretically limited to the size of the stop. Skewed distributions will be important when we try to measure the probabilities later in this cheater.

Characteristics of the Principal Averages

Each averaging method has its unique meaning and usefulness. The following summary points out their principal characteristics:

The arithmetic mean is affected by each data element equally, but it has a tendency to enphasize extreme values more than other methods. It is easily calculated and is subject to algebraic manipulation.

The geometric mean gives less weight to extreme variations than the arithmetic mean and is most important when using data representing ratios or rates of change. It cannot ahvaj-s be used for a combination of positive and negative numbers and is also subject to algebraic manipulation.

The harmonic mean is most applicable to time changes and, along with the geometric mean, has been used in economics for price analj-sis. added complications of computation have caused this to be less popular than either of the other averages, although it is also capable of algebraic manipulation.

The mode is not affected by the size of the variations from the average, only the distribution. It is the location of

greatest concentration and indicates a tjpical value for a reasonably laie sample. With an unordered set of data the mode is time consuming to locate and is not capable of algebraic manipulation.

The median is most useful when the center of an incomplete set is needed. It is not affected by extreme variations and is simple to fmd if the number of data points are known. Although it has some arithmetic properties, it is not readily adaptable to computational methods.

DISPERSION AND SKEWNESS

The center or central tendency of a data series is not a sufficient description for price analj-sis. The manner in which it is scattered about a given point, its dispersion and shewness, are necessarj to describe the data. The mean deviation is a basic method for measuring distribution and may be calculated about any measure of central location, for exanple, the arithmetic mean. It is found by computing

, lp"".-

where MD is the mean deviation, the average of the differences between each price and the arithmetic mean of the prices, or other measure of central location, with signs ignored.

The standard deviation is a special form of measuring average deviation from the mean, which uses the root-mean-square

where the differences between the individual prices and the mean are squared to emphasize the significance of extreme values, and then total final value is scaled bad; using the square root fundion. This popular measure, found throughout this book, is available in all sjreadsheets and software programs as (fflStd or (fflStdey For n prices, the standard deviation is simply (aStd(price,n).

The standard deviation is the most popular way of measuring the degree of dispersion of the data. The value of one dandard deviation about the mean represents a clustering of about 68°o of the data, two standard deviations from the mean include 95.5>o of all data, and three standard deviations encompass 99.7°o, nearly all the data. These values represent the groupings of a perfedly normal set of data, shown in Figure 2-4.

Probability of Achieving a Return

If we look at Figure 2-4 as the annual retums for the stock market over the past 50 years, then the mean is about 8% and one standard deviation is l6°o. In any one year we can exped the compounded rate of retum to be 8°o; however, there is a ilo chance that it will be either greater than 24>o (mean plus one standard deviation) or less than -8°o (the mean minus one standard deviation). If you would like to know the probability of a retum of 20°o or greater, you must first rescale the values.

FIGURE 2-4 Normal distribution showing the percentage area included within one standard deviation about the arithmetic mean.

We look in Appendix Al under the probability for normal airves, and find that a standard deviation of .75 gives 27.340o, a grouping of 54.68°o of the data. That leaves one-half of the remaining data, or 22.66°o, above the target of 20° o.

Skewness

Most price data, however, are not normally distributed. For phj-sical commodities, such as gold, grains, and interest rates (yield), prices tend to spend more time at low levels and much less time at extreme highs; while gold peaked at $800 per ounce for one day, it has remained between $375 and $400 per ounce for most of the past 10 years. The possibility of failing below $400 by the same amount as its rise to $800 is impossible, unless you believe that gold can go to zero. This relationship of price versus time, in which martlets spend more time at lower levels, can be measured as skewnessthe amount of distortion from a sj-mmetric shape that makes the curve appear to be short on one side and extended on the other, in a perfectly normal distribution, the median and mode coincide. As prices become extremely high, which often happens for short intervals of time, the mean will show the greatest change and the mode will show the least. The difference between the mean and the mode, adjusted for dispersion using the standard deviation of the distribution, gives a good measure of skewness

Because the distance between the mean and the mode, m a moderately skewed distribution, is three times the distance between the mean and the median, the relationship can also be written as:

This last formula may be more practical for computer applications, because the mode requires dividing the data into groups and counting the number of occurrences in each bar. AVhen interpreting the value of S„ the distribution leans to the right when S, is positive (the mean is greater than the median), and it is skewed left when S„ is negative.

Kunosis

One last measurement, that of kurtosis, should be familiar to analysts. Kurtosis is the 11 peakedness" of a distribution, the analj-sis of "cenfral tendency." For most cases a smaller standard deviation means that prices are clustered closer together; however, this does not ahvaj-s describe the distribution clearly Because so much of identifjing a frend comes down to deciding whether a price change is normal or likely to be a leading indicator of a new direction, deciding whether prices are closely grouped or broadly distributed may be useful. Kurtosis measures the height of the distribution.

Transformations

The diewness of a data series can sometimes be corrected using a fransformation on the data. Price data may be skewed in a specific pattem. For example, if there are 1/4 of the occurrences at twice the price and 1/9 of the occurrences at three times the price, the original data can be fransformed into a normal distribution by taking the square root of each data item. The characteriatics of price data often show a logarithmic, power, or square-root relationship.

Skewness in Price Distributions

Because the lower price levels of most commodities are determined by production costs, price distributions show a clear boundarj of resistance in that direction. At the high levels, prices can have a very long tail of low